His first success was to prove a conjecture of fellow Fields Medalist Edward Witten about the moduli space of algebraic curves. He then extended these ideas to produce many new invariants for knots and three-dimensional manifolds. He established theorems about the number of rational curves on Calabi-Yau three-manifolds that proved decisive in the development of mirror symmetry, a theory that unites methods from mathematical physics and classical algebraic geometry.

In addition to the Fields Medal, Kontsevich was the recipient of numerous other honours, including the Crafoord Prize (2008), which recognized his “important contributions to mathematics inspired by modern theoretical physics.”